a. ĐKXĐ: \(x\ne1\)
\(\dfrac{7x-1}{x-1}=\dfrac{2}{3}\)
\(\Leftrightarrow\dfrac{3\left(7x-3\right)}{3\left(x-1\right)}=\dfrac{2\left(x-1\right)}{3\left(x-1\right)}\)
\(\Rightarrow21x-9=2x-2\)
\(\Leftrightarrow21x-2x=-2+9\\ \Leftrightarrow19x=7\)
\(\Leftrightarrow x=\dfrac{7}{19}\) \(\left(TM\right)\)
Vậy phương trình có nghiệm duy nhất là \(x=\dfrac{7}{19}\)
b. ĐKXĐ: \(x\ne-1\)
\(\dfrac{2\left(3-7x\right)}{1+x}=\dfrac{1}{2}\)
\(\Leftrightarrow\dfrac{4\left(3-7x\right)}{2\left(1+x\right)}=\dfrac{1+x}{2\left(1+x\right)}\)
\(\Leftrightarrow12-28x=1+x\\ \Leftrightarrow12-1=x+28x\)
\(\Leftrightarrow11=29x\\ \Leftrightarrow x=\dfrac{11}{29}\) (TM)
Vậy phương trình có nghiệm duy nhất là \(x=\dfrac{11}{29}\)
c. ĐKXĐ: \(x\ne2\)
\(\dfrac{1}{x-2}+3=\dfrac{3-x}{x-2}\)
\(\Leftrightarrow\dfrac{1}{x-2}+\dfrac{3\left(x-2\right)}{x-2}=\dfrac{3-x}{x-2}\)
\(\Leftrightarrow1+3x-6=3-x\\ \Leftrightarrow1-6-3=-x-3x\)
\(\Leftrightarrow-8=-4x\)
\(\Leftrightarrow x=2\left(KTM\right)\)
Vậy phương trình vô nghiệm
d. ĐKXĐ: \(x\ne7\)
\(\dfrac{8-x}{x-7}-8=\dfrac{1}{x-7}\)
\(\Leftrightarrow\dfrac{8-x}{x-7}-\dfrac{8\left(x-7\right)}{x-7}=\dfrac{1}{x-7}\)
\(\Leftrightarrow8-x-8x+56=1\)
\(\Leftrightarrow8+56-1=x+8x\\ \Leftrightarrow63=9x\)
\(\Leftrightarrow x=7\left(KTM\right)\)
Vậy pt vô nghiệm
e. ĐKXĐ: \(x\ne\pm5\)
\(\dfrac{x+5}{x-5}-\dfrac{x-5}{x+5}=\dfrac{20}{x^2-25}\)
\(\Leftrightarrow\dfrac{\left(x+5\right)^2}{x^2-25}-\dfrac{\left(x-5\right)^2}{x^2-25}=\dfrac{20}{x^2-25}\)
\(\Leftrightarrow x^2+10x+25-x^2+10x-25=20\)
\(\Leftrightarrow20x=20\)
\(\Leftrightarrow x=1\left(TM\right)\)
Vậy pt có nghiệm duy nhất \(x=1\)
f. ĐKXĐ: \(x\ne\pm1\)
\(\dfrac{1}{x-1}+\dfrac{2}{x+1}=\dfrac{2}{x^2-1}\)
\(\Leftrightarrow\dfrac{x+1}{x^2-1}+\dfrac{2\left(x-1\right)}{x^2-1}\)\(=\dfrac{x}{x^2-1}\)
\(\Rightarrow x+1+2x-1=x\)
\(\Leftrightarrow x+2x-x=1-1\\ \Leftrightarrow2x=0\)
\(\Leftrightarrow x=0\left(TM\right)\)
Vậy pt có nghiệm duy nhất \(x=0\)
